Deterministic quantum partial search for target states with proportion 1/16

Abstract

The Grover search algorithm provides a quadratic speedup for searching unstructured databases. Despite its significant theoretical advantage, the algorithm can only deterministically find the target item when the proportion of target states is $\frac{1}{4}$. Grover and Radhakrishnan introduced the quantum partial search algorithm, in which information is encoded using two blocks of qubits. In this paper, we investigate the conditions that enable a deterministic quantum partial search algorithm. We demonstrate that, under two conditions: (i) the target block proportion is $\frac{1}{4}$, and (ii) the proportion of target states within the block is also $\frac{1}{4}$ (hence the proportion of target states is $\frac{1}{16}$), deterministic searching can be achieved with a sequence of operations $G_m{G}_n{G}_m^2$, requiring only one additional oracle call beyond the optimal number value of classical one. Specifically, for a database of size $2^n$ and $2^{n-4}$ target states, we identify $n-3$ partition combinations that can ensure the realization of deterministic quantum partial search. This approach not only generalizes Grover's algorithm but also provides a practical method for efficiently searching large, unstructured databases when the target state density is low.